Distributionally robust portfolio maximisation and marginal utility pricing in one period financial markets

Abstract: We consider the optimal investment and marginal utility pricing problem of a risk averse agent and quantify their exposure to a small amount of model uncertainty. Specifically, we compute explicitly the first-order sensitivity of their value function, optimal investment policy and marginal option prices to model uncertainty. The latter is understood as replacing a baseline model $\mathbb{P}$ with an adverse choice from a small Wasserstein ball around $\mathbb{P}$ in the space of probability measures. Our sensitivities are thus fully non-parametric. We show that the results entangle the baseline model specification and the agent's risk attitudes. The sensitivities can behave in a non-monotone way as a function of the baseline model's Sharpe's ratio, the relative weighting of assets in an agent's portfolio can change and marginal prices can increase when an agent faces model uncertainty.